Your proof and the proof from the book are equally correct. I like your proof because it doesn't rely on that theorem.
Is this the correct way to prove (-a)b=-ab?
Since -a+a=0, then (-a)b+ab=0b, so (-a)b+ab=0. Then (-a)b=-ab.
I was wondering because the book I have uses the fact that -ab+ab=0 to create (-a)b+ab=-ab+ab and uses a theorem, a+c=b+c then a=b, to show (-a)b=-ab.